Tangentially, to some extent.
Invisibility is a spell effect (usually) and spell rules are pretty hard and fast and more or less always have been.
But take lockpicking. Time was, any character could try this and not necessarily auto-fail but a few classes were clearly better at it. Eventually, that morphed into don't even try unless you've got the skill or ability as you'll auto-fail every time.
To be fair, 5e has rolled this back a lot. Technically anyone can try to lockpick, and even more technically, you don't even need lockpicks to try it. IT is the difference between:
Roll mod with disadvantage -> Roll Mod -> Roll Mod with Proficiency -> Roll Mod with Expertise
The issue (if you can even call it that) is that a lot of players recognize this is a team game. And that means that while, yes, the character with no proficiency could absolutely try, there is a guy who got lockpicks and has proficiency, so he should be the one to try it. This is compounded by the fact of multiple resolution paths, but only allowing a single binary success/fail. If you have the choice between "rolling mod with disadvantage" for trying to lockpick with a hairpin which is silent or "roll mod with proficiency" for shattering the door with your mighty thews... well, you know that you aren't likely to succeed the first way, so you ignore it and move on to how you
can succeed.
These issues combine to create the reality of the game, where no one really tries anything unless they are the one proficient and with the highest mod, because success is binary, and you don't want to be the one who fails on an important point.
I also want to rollback and address something else. The fact that DnD is a team game really isn't thought about enough in terms of the fiction we reference. Conan, Elric, ect they may have "companions" but how often are the heroes left to do everything except one or two specific tasks? Think for a moment about Bilbo being a thief. With both Gollum and with Smaug (two of the best scenes) Bilbo is alone. Same with the spiders in the forest or the Trolls, the dwarves had been kidnapped and he was rescuing them.
This leads us, I think, to want our heroes to be incredibly versatile. You should be able to talk and steal and fight and hide and run and maybe even do magic. But, if you try and build that sort of character, unless you get very lucky, you are just second string to the others who are specialists. And since each scene is really being conceived as a challenge one person can do, then it leads to this weird space where only one or two characters participate while the others (much like the dwarves vis a vis Smaug) are left waiting outside the Mountain.
And interestingly, this also explains why everyone is good at combat, because everyone is forced to participate in combat, and everyone had to get good at it, in the course of the game's design, because while the theif might go and unlock doors alone, no one is expected to fight solo while the rest of the party watches.